Leibniz had a vision of a universal ambiguity-free language based on a new symbol set, a characterica universalis, and a machine-like calculus ratiocinator that would automatically prove all necessary truths, true in "all possible worlds." Gottlob Frege called the idea "a system of notation directly appropriate to objects." In the three hundred years since Leibniz had this vision, logical philosophers and linguistic analysts have sought those truths in the form of "truth-functional" propositions and statements formulated in words, but they have failed to find a necessarily "true" connection between words and objects.

Information philosophy uses such system of notation, not in words, but in bits of digital information. And the interconnected computers of the Internet are not only Leibniz's calculus ratiocinator, but humanity's storehouse of shared experiences and accumulated knowledge. Like the individual Experience Recorder and Reproducer (ERR) in each human mind, the World Wide Web is our shared Knowledge Recorder and Reproducer. Computer simulations of physical and biological processes are the best representations of human knowledge about the external world of objects.

Leibniz's Principle of Sufficient Reason says that every event has a reason or cause in the prior state of the world. This appears to commit him to a necessarydeterminism, but like the ancient compatibilistChrysippus, Leibniz argues that some empirical things are contingent.

Another of his great principles is his Principle of Contradiction (Aristotle's Principle of Non-Contradiction), a proposition cannot be true and false at the same time, and that therefore A is A and cannot be not A.

That "A is A" follows from what Leibniz called the Identity of Indiscernibles, which came to be known as Leibniz's Law. In modern times, philosophers use both the Identity of Indiscernibles and the converse Indiscernibility of Identicals, often calling either of these Leibniz's Law.

We must distinguish between these concepts, first by noting that given two numerically distinct things that are "identical," by definition there can be no discernible differences between them. This Indiscernibility of Identicals may be simply an ideal concept.

Leibniz says that given the apparent indiscernibility of two things, they are nevertheless not identical if there are differences between them beyond the reach of our senses, differences too small to be discernible.

Leibniz himself described this possibility. And he was very clear that even if minute differences are indiscernible so that two distinct things appear to be identical, there are simply no two things perfectly identical that only differ in number (solo numero).

Leibniz thought that perfect identity can only be the identity of one thing with itself. Two distinct things that are indiscernible must be same thing under two names, he said (anticipating the modern discussions starting with Gottlob Frege about the Morning Star and Evening Star).

Information philosophy now shows that two concrete objects or abstract entities can be perfectly identical if and only if it is their intrinsic properties are compared. If their relational properties with other objects and position in space and time are ignored, two objects may be intrinsic information identicals.

Despite his denial of necessarily identical things in the world, Leibniz was a great mathematician and articulated a Principle of Substitutivity, that identical things can be substituted for one another without changing the meaning or truth value of a statement. Gottlob Frege, Bertrand Russell, and Ludwig Wittgenstein all followed Leibniz and accepted this principle of substitutivity.

The Metaphysics of Identity

Leibniz calls the identity of any object with itself as a primary truth.

Primary truths are those which either state a term of itself or
deny an opposite of its opposite. For example, 'A is A', or
'A is not not-A'; If it is true that A is B, it is false that A is
not B, or that A is not-B'; again, 'Each thing is what it is',
'Each thing is like itself, or is equal to itself, 'Nothing is
greater or less than itself—and others of this sort which,
though they may have their own grades of priority, can all be
included under the one name of 'identities'.

4. There are no two individuals indiscernible from one
another... Two drops of water or milk
looked at under the microscope will be found to be discernible.
This is an argument against atoms, which, like the void, are
opposed to the principles of a true metaphysic.

The Identity of Indiscernibles is known as Leibniz's Law

Cf., Hesperus and Phosphorus as identical to Venus

5. These great principles of a Sufficient Reason and of the
Identity of Indiscernibles change the state of metaphysics,
which by their means becomes real and demonstrative;
whereas formerly it practically consisted of nothing but empty
terms.

6. To suppose two things indiscernible is to suppose the
same thing under two names. [Cf. Frege]

In a word, the insensible perception is of as much use in
pneumatics as is the insensible corpuscle in physics; and it
is equally unreasonable to reject the one or the other on the
pretext that it is beyond the reach of our senses...

To think otherwise
is to have but little knowledge of the immensely subtle
composition of things, which always and everywhere include
an actual infinity.

I have also noted that, by virtue of insensible variations,
two individual things can never be perfectly alike, and that
they must always differ more than numero...

if we thought in good
earnest that the things we do not apperceive are not there in
the soul or in the body, we should fail in philosophy as in
politics, by neglecting το μικρóν insensible progressions;
whereas an abstraction is not an error, provided we know that
what we are ignoring is really there. This is the use made
of abstractions by mathematicians when they speak of the
perfect lines they ask us to consider, and of uniform motions
and other regular effects, although matter (that is to say the
mixture of the effects of the surrounding infinite) is always
providing some exception. We proceed in this way so as to
distinguish the various considerations from one another, and
to reduce the effects to their reasons as far as is possible to us,
and to foresee some consequences; for the more careful we are
to neglect no consideration which we can regulate, the more
does practice correspond to theory.

This knowledge of insensible perceptions serves also to
explain why and how two souls, whether human or of some
other identical species, never come perfectly alike from the
Creator's hands, but each has always from the beginning its
own relation to the point of view it will have in the universe.
But this follows from what I pointed out previously about
two individuals, namely that their difference is always more than a numerical one.

(New Essays on the HumanUnderstanding, p.56)

In his Monadology, Leibniz maintains that even though his Monads are the simplest of all substance, with no internal parts, they are nevertheless numerically distinct individuals, because there are never in nature two things exactly alike.

1. The Monad, of which we will speak here, is
nothing else than a simple substance, which goes to
make up composites; by simple, we mean without
parts.

2. There must be simple substances because there
are composites; for a composite is nothing else than
a collection or aggregatum of simple substances.

3. Now, where there are no constituent parts
there is possible neither extension, nor form, nor
divisibility. These Monads are the true Atoms of
nature, and, in fact, the Elements of things...

9. Each Monad, indeed, must be different from
every other. For there are never in nature two
beings which are exactly alike, and in which it is
not possible to find a difference either internal or
based on an intrinsic property.

(Monadology)

Leibniz's emphasis on intrinsic, internal properties is very insightful. In his posthumous New Essays responding to Locke, clearly says that extrinsic properties like space and time, and the relations to other things outside allow us to distinguish things we might not be able to tell apart by reference to themselves alone.

He also says two things cannot occupy the same place and time.

Chapter xxvii
What identity or diversity is

PHILALETHES. §1. A relative idea of the greatest importance is that of
identity or of diversity. We never find, nor can we conceive it possible,
that two things of the same kind should exist in the same place at the same
time. That is why, when we demand, whether any thing be the same or
no, it refers always to something that existed such a time in such a place
. . . . From whence it follows, that one thing cannot have two beginnings of
existence, nor two things one beginning. . .in time and place'.

THEOPHILUS. In addition to the difference of time or of place there must
always be an internal principle of distinction: although there can be many
things of the same kind, it is still the case that none of them are ever exactly
alike. Thus, although time and place (i.e. the relations to what lies outside)
do distinguish for us things which we could not easily tell apart by
reference to themselves alone, things are nevertheless distinguishable in
themselves. So time and place do not constitute the core of identity and
diversity, despite the fact that diversity in time or place brings with it
differences in the states that are impressed upon a thing, and thus goes
hand in hand with diversity of things. To which it can be added that it is
by means of things that we must distinguish one time or place from
another, rather than vice versa; for times and places are in themselves
perfectly alike, and in any case they are not substances or complete
realities. The method which you seem to be offering here as the only one
for distinguishing among things of the same kind, is founded on the
assumption that interpenetration is contrary to nature. This is a reasonable
assumption; but experience itself shows that we are not bound to it when
it comes to distinguishing things. For instance, we find that two shadows
or two rays of light interpenetrate, and we could devise an imaginary
world where bodies did the same. Yet we can still distinguish one ray from
the other just by the direction of their paths, even when they intersect.

P H I L . §3. What is called the principle of individuation in the Schools, where
it is so much inquired after,... is existence it self, which determines a
being. . .to a particular time and place incommunicable to two beings of
the same kind.'

THEO. The 'principle of individuation' reduces, in the case of individuals,
to the principle of distinction of which I have just been speaking. If two
individuals were perfectly similar and equal and, in short, indistinguishable
in themselves, there would be no principle of individuation. I would even
venture to say that in such a case there would be no individual distinctness,
no separate individuals. That is why the notion of atoms is chimerical and
arises only from men's incomplete conceptions. For if there were atoms,
i.e. perfectly hard and perfectly unalterable bodies which were incapable
of internal change and could differ from one another only in size and in
shape, it is obvious that since they could have the same size and shape they
would then be indistinguishable in themselves and discernible only by
means of external denominations with no internal foundation; which is
contrary to the greatest principles of reason. In fact, however, every body
is changeable and indeed is actually changing all the time, so that it differs
in itself from every other. I remember a great princess [*Sophie], of lofty
intelligence, saying one day while walking in her garden that she did not
believe there were two leaves perfectly alike. A clever gentleman who was
walking with her believed that it would be easy to find some, but search
as he might he became convinced by his own eyes that a difference could
always be found. One can see from these considerations, which have until
now been overlooked, how far people have strayed in philosophy from the
most natural notions, and at what a distance from the great principles of
true metaphysics they have come to be.

(New Essays on the HumanUnderstanding, pp.229-231)

Necessary and Contingent Truths

Leibniz knows that necessary truths are those where the information in the predicate is already in the subject. These are analytical statements whose opposite implies a contradiction, which can be resolved into identity statements. Such propositions that express necessary truths ("true in all possible worlds) are now seen to be tautological and carrying no new information. Yet Leibniz here talks as if they contain knowledge of the essence and existence of things.

True in all possible worlds simply means that their truth is independent of the physical world. Kant will describe such statements as analytic.

An affirmative truth is one whose predicate is in the subject;
and so in every true affirmative proposition, necessary or contingent,
universal or particular, the notion of the predicate
is in some way contained in the notion of the subject, in such
a way that if anyone were to understand perfectly each of the
two notions just as God understands it, he would by that
very fact perceive that the predicate is in the subject. From
this it follows that all the knowledge of propositions which is in
God, whether this is of the simple intelligence, concerning
the essence of things, or of vision, concerning the existence
of things, or mediate knowledge concerning conditioned
existences, results immediately from the perfect understanding
of each term which can be the subject or predicate
of any proposition. That is, the a priori knowledge of complexes
arises from the understanding of that which is not
complex.

An absolutely necessary proposition is one which can be resolved
into identical propositions, or, whose opposite implies a contradiction...
This type of necessity,
therefore, I call metaphysical or geometrical. That which
lacks such necessity I call contingent, but that which implies a
contradiction, or whose opposite is necessary, is called impossible.
The rest are called possible.

In the case of a contingent truth, even though the predicate
is really in the subject, yet one never arrives at a
demonstration or an identity, even though the resolution of
each term is continued indefinitely. In such cases it is only
God, who comprehends the infinite at once, who can see how
the one is in the other, and can understand a priori the perfect
reason for contingency; in creatures this is supplied a posteriori,
by experience. So the relation of contingent to necessary
truths is somewhat like the relation of surd ratios (namely, the
ratios of incommensurable numbers) to the expressible ratios
of commensurable numbers.

Leibniz knew that our ideas, even those we think necessary, began with our sensory experiences. His famous idea that some "necessary truths" are "true in all possible worlds," was more fundamentally his insight that the necessary truths do not depend in any way on the physical world or our senses.

From this arises another question, whether all
truths depend on experience, that is to say on induction and on
instances, or whether there are some which have another
basis also. For if certain events can be foreseen before we have
made any trial of them, it is clear that we contribute in those
cases something of our own. The senses, although they are
necessary for all our actual knowledge, are not sufficient to
give us the whole of it, since the senses never give anything
but instances, that is to say particular or individual truths.
Now all the instances which confirm a general truth, however
numerous they may be, are not sufficient to establish
the universal necessity of this same truth, for it does not follow
that what happened before will happen in the same way
again... And any one
who believed that [day must follow night] is a necessary and
eternal truth which will last for ever, would likewise be wrong,
since we must hold that the earth and even the sun do not
exist of necessity, and that there may perhaps come a time
when that beautiful star and its whole system will exist no
longer, at least in its present form.

The idea of truths necessary "in all possible worlds" is really that "proofs" do not depend on instances, sensory evidence. Necessity is independent of the physical world

From which it appears that
necessary truths, such as we find in pure mathematics, and
particularly in arithmetic and geometry, must have principles
whose proof does not depend on instances, nor consequently
on the testimony of the senses, although without the
senses it would never have occurred to us to think of them.

(New Essays on the Human Understanding, Preface, p.49)

The Scholastics Right about Immaterial Forms

11. That the meditations of the so-called Scholastic philosophers and theologians are not to be entirely despised.

I know that I am putting forward a great paradox in claiming
to rehabilitate ancient philosophy to some extent, and to
restore the rights of citizenship to substantial forms, which have
practically been banished. But perhaps I shall not readily
be condemned when it is known that I have thought carefully
about modern philosophy, and that I have devoted much
time to physical experiments and to geometrical demonstrations. I was for a long time persuaded of the emptiness of
these entities, and was finally obliged to take them up again
despite myself, and as it were by force. This was after I had
myself conducted some researches which made me recognise
that our modern philosophers do not do enough justice to St.
Thomas and to other great men of that era, and that the
views of the Scholastic philosophers and theologians have
much mere soundness than is imagined, provided that one
uses them in a proper way and in their right place. I am even
persuaded that if some, precise and thoughtful mind were to
take the trouble of clarifying and setting in order their
thoughts, in the manner of analytic geometry, he would find
in them a treasury of truths which are extremely important
and wholly demonstrative.

12.That the notions which consist in extension include something imaginary,
and cannot constitute the substance of a body.

But to take up again the thread of our discussion: I believe
that anyone who will meditate on the nature of substance, as
I have explained it above, will find that the entire nature of
body does not consist in extension alone, that is to say in size,
shape and motion. Rather, he will find that it is necessary to
recognise in it something which has some relation to souls,
and which is comrnonly called a substantial form, although
this changes nothing in phenomena, any more than the soul
of the lower animals does, if they have one. It can even be
demonstrated that the notion of size, shape and motion is not
as distinct as is imagined, and that it contains something that
is imaginary and relative to our perceptions, just as is the
case (though even more so) with colour, heat and other
similar qualities, of which it may be doubted whether they
are really found in the nature of things outside us. This is why
qualities of these kinds cannot constitute any substance. And
if there is no other principle of identity in bodies besides that
which we have just mentioned, no body will ever last longer
than a moment. However, the souls and substantial forms of
other bodies are very different from intelligent souls. Only
the latter know their actions, and not only do not perish
naturally, but even retain perpetually the basis of the knowledge of what they are. It is this which brings it about that
they alone are capable of punishment and reward, and makes
them citizens of the commonwealth of the universe, of which
God is the monarch. It also follows that all other creatures
must serve them, of which we shall speak at greater length
presently.

It is a very old doubt of mankind, how freedom and contingency
can be reconciled with the series of causes and with
providence. The difficulty of the matter has been increased
by the dissertations of Christian authors on God's justice in
procuring the salvation of men.

For my part, I used to consider that nothing happens by
chance or by accident, except with respect to certain particular
substances; that fortune, as distinct from fate, is an
empty word; and that nothing exists unless its individual
requisites are given, and that from all these taken together
it follows that the thing exists. So I was not far from the view
of those who think that all things are absolutely necessary;
who think that security from compulsion is enough for freedom,
even though it is under the rule of necessity, and who
do not distinguish the infallible—that is, a truth which is
certainly known—from the necessary.

But I was dragged back from this precipice by a consideration
of those possibles which neither do exist, nor will exist,
nor have existed. For if certain possibles never exist, then
existing things are not always necessary; otherwise it would
be impossible for other things to exist instead of them, and so
all things that never exist would be impossible. For it cannot
be denied that many stories, especially those which are called
'romances', are possible, even if they do not find any place in
this series of the universe, which God has chosen—unless
someone supposes that in the vast magnitude of space and
time there exist the regions of the poets, where you could see
wandering through the world King Arthur of Britain, Amadis
of Gaul, and Dietrich von Bern, famed in the stories of the
Germans. A certain distinguished philosopher of our century [Spinoza]
seems to have been close to this opinion, for he says expressly
somewhere that matter takes on successively all the forms of
which it is capable (Principles of Philosophy, Part III, art. 47).
This view is indefensible, for it would remove all the beauty
of the universe and all choice, to say nothing here of other
arguments by which the contrary can be shown.

Once I had recognised the contingency of things, I then
began to consider what a clear notion of truth would be; for
I hoped, not unreasonably, to derive from this some light
on the problem of distinguishing necessary from contingent
truths. However, I saw that it is common to every true
affirmative proposition—universal and particular, necessary
or contingent—that the predicate is in the subject, or that the
notion of the predicate is in some way involved in the notion
of the subject, and that this is the principle of infallibility
in every kind of truth for him who knows everything a priori.
But this seemed to increase the difficulty. For if, at a given
time, the notion of the predicate is in the notion of the subject,
then how, without contradiction and impossibility, can
the predicate not be in the subject at that time, without destroying
the notion of the subject?

A new and unexpected light finally arose in a quarter where
I least hoped for it—namely, out of mathematical considerations
of the nature of the infinite. There are two labyrinths
of the human mind: one concerns the composition of the
continuum, and the other the nature of freedom, and both
spring from the same source—the infinite. That distinguished
philosopher whom I mentioned above could not unravel
these knots, or at any rate was unwilling to make his opinion
known, but preferred to cut them with a sword. For he says
(Principles of Philosophy, Part I, arts. 40 and 41) that we can
easily involve ourselves in great difficulties if we try to reconcile
God's preordination with the freedom of the will, and
that we must abstain from discussing them, since God's
nature cannot be comprehended by us. He also says (Part II.
art. 35) that we ought not to doubt that matter is divided ad
infinitum, even though we cannot understand this. But tl-.b
is not enough: for it is one thing for us not to understand a
thing, and another for us to understand its contradictory.

So it is at all events necessary to be able to answer those
arguments which seem to imply that freedom or the division
of matter imply a contradiction.

It must be known, therefore, that all creatures have impressed
on them a certain mark of the divine infinity, and
that this is the source of many wonders which amaze the
human mind.

For example, there is no portion of matter so small that
there does not exist in it a world of creatures, infinite in
number. Again?"every individual created substance, however
imperfect, acts on all others and is acted on by all others,
and contains in its complete notion (as this exists in the mind
of God) the whole universe, and whatever is, was or will be.
Further, every truth of fact or of individual things depends
on a series of infinite reasons, and all that is in this series
can be seen by God alone. This is also the reason why God
alone knows contingent truths a priori, and sees their infallibility
in another way than by experience.

When I had considered these more attentively, a profound
difference between necessary and contingent truths
came to light. Every truth is either original or derivative.
Original truths are those of which a reason cannot be given;
such truths are identical or immediate, and they affirm a term
of itself or deny a contradictory of its contradictory."1 Derivative
truths are again of two sorts: some are analysed into
original truths, others admit of an infinite process of analysis.
The former are necessary, the latter contingent. A necessary
proposition is one whose contrary implies a contradiction,
such as all identical propositions and all derivative propositions
which are analysable into identical propositions. These are the
truths which are said to be of metaphysical or geometrical
necessity. For demonstration consists simply in this: by the
analysis of the terms of a proposition, and by substituting for
a defined term a definition or part of a definition, one shows a
certain equation or coincidence of predicate with subject in a
reciprocal proposition, or in other cases at least the inclusion
of the predicate in the subject, in such a way that what was
latent in the proposition and as it were contained in it virtually
is rendered evident and express by the demonstration...

But in the case of contingent truths, even though the predicate
is in the subject, this can never be demonstrated of it,
nor can the proposition ever be reduced to an equation or
identity. Instead, the analysis proceeds to infinity, God alone
seeing—not, indeed, the end of the analysis, since it has no
end—but the connexion of terms or the inclusion of the predicate
in the subject, for he sees whatever is in the series;
indeed, this very same truth has arisen in part from his own
intellect and in part from his will, and expresses in its own
way his infinite perfection and the harmony of the whole
series of things.

However, there have been left to us two ways of knowing
contingent truths; one is the way of experience and the other
the way of reason. The way of experience is when we perceive
a thing clearly enough by our senses; the way of reason
is derived from the general principle that nothing happens
without a reason, or, that the predicate is always in some way
in the subject.

Leibniz imagined a scientist who could see the events of all times and predict the future, just as all times are thought to be present to the mind of God.

"Everything proceeds mathematically...if someone could have a sufficient insight into the inner parts of things, and in addition had remembrance and intelligence enough to consider all the circumstances and take them into account, he would be a prophet and see the future in the present as in a mirror."

Pierre-Simon Laplace particularized this Leibniz vision as an intelligent being who knows the positions and velocities of all the atoms in the universe and uses Newton's equations of motion to predict the future. Laplace's Demon has become a cliché for physical determinism.